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arXiv · 1604.05884

Eigenvalues of Weighted-Laplacian under the extended Ricci flow

Abimbola Abolarinwa
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Abstract

Let $\Delta_\varphi = \Delta -\nabla \varphi \nabla$ be a symmetric diffusion operator with an invariant weighted volume measure $d\mu = e^{-\varphi} dv$ on an $n$-dimensional compact Riemannian manifold $(M,g)$, where $g=g(t)$ solves the extended Ricci flow. In this article we study the evolution and monotonicty of the first nonzero eigenvalue of $\Delta_\varphi$ and we obatin several monotone quantities along the extended Ricci flow and its volume preserving version under some technical assumption. We also show that the eigenvalues diverge in a finite time for the case $n\geq 3$. Our results are natural extension of some known results for Laplace-Beltrami operator under various geometric flows.

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Abimbola Abolarinwa. 2016-04-20. Eigenvalues of Weighted-Laplacian under the extended Ricci flow. https://arxiv.org/abs/1604.05884

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